Cartan subalgebra

Cartan subalgebra
Jgmoxness · CC BY-SA 3.0 · source
Name	Cartan subalgebra
Field	Lie theory
Introduced	Élie Cartan
Notable contributors	Élie Cartan, Hermann Weyl, Claude Chevalley, Nathan Jacobson, Armand Borel

Cartan subalgebra

A Cartan subalgebra is a maximal nilpotent or maximal toral self-normalizing subalgebra arising in the theory of Lie algebras and Lie groups, central to structural classification, representation theory, and geometry. The notion was developed in the work of Élie Cartan and later refined by Hermann Weyl, Claude Chevalley, Armand Borel, and Nathan Jacobson, linking algebraic groups, symmetric spaces, and the theory of roots and weights. Cartan subalgebras serve as a pivot between global objects such as Lie groups, algebraic groups like GL_n(C), and classical structures related to SO_n, SL_n, and exceptional groups like E8.

Definition and basic properties

In the context of a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, a Cartan subalgebra is defined as a nilpotent subalgebra that equals its normalizer and is maximal with this property, a notion shaped by the studies of Élie Cartan and Hermann Weyl. For complex semisimple Lie algebras studied by Claude Chevalley and Armand Borel, Cartan subalgebras are precisely maximal toral (diagonalizable) subalgebras, linking to classical works on Cartan decomposition and Weyl character formula by Hermann Weyl. Important properties include conjugacy under the action of automorphism groups like Ad(G) for connected groups such as SL_n(C), and the fact that their dimension equals the rank of the ambient algebra, a notion used in the classification by Elie Cartan and later by Nathan Jacobson.

Examples and classification

Canonical examples include the diagonal matrices in gl_n(C) and sl_n(C), used extensively in the classification of classical Lie algebras such as so_n(C) and sp_{2n}(C), and appearing in the study of groups like GL_n(R), SU(n), and Sp(n). Exceptional cases arise in the exceptional Lie algebras G2, F4, E6, E7, and E8 that were catalogued following the Cartan–Killing classification and influenced the work of Élie Cartan and Wilhelm Killing. Over nonalgebraically closed fields or fields of positive characteristic treated in Chevalley group theory, Cartan subalgebras may have subtler forms as explored by Serre and J.-P. Serre in his treatment of algebraic groups and reductive groups. In matrix Lie algebras appearing in the literature of Sophus Lie and later texts by Jacobson and Bourbaki, explicit Cartan subalgebras provide concrete root systems and weight spaces.

Cartan subalgebras in semisimple Lie algebras

For semisimple Lie algebras, the Cartan subalgebra is central to structure theory used by Élie Cartan, Hermann Weyl, and Claude Chevalley to derive root systems and the Weyl group; it gives rise to the root space decomposition underpinning the Cartan–Weyl basis and the Killing form-based criteria developed by Wilhelm Killing and Élie Cartan. Conjugacy of Cartan subalgebras under the connected algebraic group such as Int(g) or classical groups like SO_n(C) and Sp_{2n}(C) is a foundational theorem proved in the classical period and refined in texts by Nathan Jacobson and Armand Borel. The Cartan subalgebra determines the Dynkin diagram classification of semisimple Lie algebras completed in the works of Killing and Élie Cartan, later formalized in modern expositions by Bourbaki.

Cartan subalgebras in solvable and nilpotent Lie algebras

In solvable and nilpotent Lie algebras, as investigated by Nathan Jacobson and in the development of Lie's theorem by Sophus Lie, Cartan subalgebras need not be unique up to conjugacy, and their structure is more intricate than in the semisimple case. For nilpotent Lie algebras studied in classifications related to Heisenberg groups and nilpotent orbits examined by Joseph and Dixmier, Cartan subalgebras coincide with maximal nilpotent self-normalizing subalgebras and are related to polarizations used in induction constructions for representations of groups treated by Mackey. Over fields of positive characteristic, researchers including Chevalley and Serre identified pathologies and modifications necessary for the notion to behave analogously to the classical setting.

Weyl group and root space decomposition

Associated to a Cartan subalgebra in a semisimple Lie algebra is the root system and the Weyl group, concepts introduced by Hermann Weyl and refined by Claude Chevalley and Armand Borel. The root space decomposition partitions the algebra into eigenspaces for the adjoint action of the Cartan subalgebra, leading to the classification via Dynkin diagrams credited to Élie Cartan and Wilhelm Killing. The Weyl group acts by reflections on the Cartan subalgebra, playing a central role in the formulation of the Weyl character formula and connections to representation theory developed by Hermann Weyl, Harish-Chandra, and George Mackey.

Construction and existence theorems

Existence and conjugacy theorems for Cartan subalgebras were established early in the 20th century by Élie Cartan and formalized in algebraic language by Nathan Jacobson and later by the collective works compiled in Bourbaki. Over algebraically closed fields of characteristic zero, Cartan subalgebras exist in every finite-dimensional Lie algebra and any two Cartan subalgebras of a semisimple Lie algebra are conjugate under automorphisms of the algebra, results that underpin constructions in the representation theory of compact Lie groups such as SU(2) and algebraic groups like SL_n(C). Extensions and refinements for algebraic groups and groups over nonclosed fields feature in work by Chevalley and Serre, and applications span to areas linked with Root system combinatorics, the theory of Kac–Moody algebras, and geometric representation theory as developed by George Lusztig and Victor Kac.

Category:Lie algebras